Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations

The bi-Hamiltonian structure of certain multi-component integrable systems, generalizations of the dispersionless Toda hierarchy, is studies for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e. the metric which defines the Hamiltonian structure has vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extanded to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, are given.Comment: 28 pages, LaTe

Multicomponent bi-superHamiltonian KdV systems

It is shown that a new class of classical multicomponent super KdV equations is bi-superHamiltonian by extending the method for the verification of graded Jacobi identity. The multicomponent extension of super mKdV equations is obtained by using the super Miura transformation

On the B\"acklund Transformation for the Moyal Korteweg-de Vries Hierarchy

We study the B\"acklund symmetry for the Moyal Korteweg-de Vries (KdV) hierarchy based on the Kuperschmidt-Wilson Theorem associated with second Gelfand-Dickey structure with respect to the Moyal bracket, which generalizes the result of Adler for the ordinary KdV.Comment: 9 pages, Revte

Conformal Covariantization of Moyal-Lax Operators

A covariant approach to the conformal property associated with Moyal-Lax operators is given. By identifying the conformal covariance with the second Gelfand-Dickey flow, we covariantize Moyal-Lax operators to construct the primary fields of one-parameter deformation of classical $W$-algebras.Comment: 13 pages, Revtex, no figures, v.2: typos corrected, references added and conclusion modifie

Dispersionful analogues of Benney's equations and NN-wave systems

We recall Krichever's construction of additional flows to Benney's hierarchy, attached to poles at finite distance of the Lax operator. Then we construct a ``dispersionful'' analogue of this hierarchy, in which the role of poles at finite distance is played by Miura fields. We connect this hierarchy with $N$-wave systems, and prove several facts about the latter (Lax representation, Chern-Simons-type Lagrangian, connection with Liouville equation, $\tau$-functions).Comment: 12 pages, latex, no figure

Quantum Sturm-Liouville Equation, Quantum Resolvent, Quantum Integrals, and Quantum KdV : the Fast Decrease Case

We construct quantum operators solving the quantum versions of the Sturm-Liouville equation and the resolvent equation, and show the existence of conserved currents. The construction depends on the following input data: the basic quantum field $O(k)$ and the regularization .Comment: minor correction

Hydrodynamic chains and a classification of their Poisson brackets

Necessary and sufficient conditions for an existence of the Poisson brackets significantly simplify in the Liouville coordinates. The corresponding equations can be integrated. Thus, a description of local Hamiltonian structures is a first step in a description of integrable hydrodynamic chains. The concept of $M$ Poisson bracket is introduced. Several new Poisson brackets are presented

h-deformation of Gr(2)

The $h$-deformation of functions on the Grassmann matrix group $Gr(2)$ is presented via a contraction of $Gr_q(2)$. As an interesting point, we have seen that, in the case of the $h$-deformation, both R-matrices of $GL_h(2)$ and $Gr_h(2)$ are the same

A Note on the Gauge Equivalence between the Manin-Radul and Laberge-Mathieu Super KdV Hierarchies

The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies is revisited. Apart from the Inami-Kanno transformation, we show that there is another gauge transformation which also possess the canonical property. We explore the relationship of these two gauge transformations from the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the Darboux-Backlund transformation for the Manin-Radul super KdV hierarchy. The geometrical intepretation of these transformations is also briefly discussed.Comment: 8 pages, revtex, 1 figur

S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.Comment: 35 pages, uses AMS-Latex, Hyperref, Geometry, Array and Babel package